Solutions to week 119

Solar Cap Versus Polar Cap answer explained:
Let the compass radius be r. On a sphere of radius R the area of a spherical cap of half-angle θ is 2π(1 – cosθ)R2 (θ is shown in the figure.) Also from the figure, cosθ = 1 – 2sin2(θ/2) = 1 – 2r2/(2R)2. Putting these together gives the area of the spherical cap as πr2, independent of R.